On the full automorphism groups of $2$-designs constructed from finite fields ${\mathbb F}_{2^n}$

TL;DR

The paper addresses the automorphism structure of two $2$-designs constructed from algebraic curves over the finite field $\mathbb{F}_q$ with $q=2^n$. By defining parabolas $U_a^q$ and hyperbolas $O_a^q$, it shows that the full automorphism groups of the resulting designs are $\mathrm{GL}_n(2)$, acting primitively on both points and blocks, and embeds these groups as primitive maximal subgroups of $\mathrm{Alt}(\mathbb{F}_q^\times)$. The work establishes a dual relationship between the designs, derives the stabilizers as $2^{n-1}:\mathrm{GL}_{n-1}(2)$, and analyzes cases $n=2$ and $n\ge3$ with attention to conjugacy and normalizer structures involving a Frobenius map and a Singer torus. It also discusses the classical nature of the parabola-based design in contrast to the non-classical hyperbola design and concludes with open questions about intersections and conjugacy within the symmetric group. Overall, it bridges finite field geometry with group actions to classify highly symmetric combinatorial structures.

Abstract

In this manuscript, for $q:=2^n$ with $n\geq2$, we study two primitive maximal subgroups of the alternating group ${\sf A}_{q-1}$. These subgroups are the full automorphism groups of $2$-designs which are constructed from algebraic curves over the finite field ${\mathbb F}_q$.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 2.1
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Theorem 3.5
• proof
• Definition 3.8
• Lemma 3.9